Schrödinger’s Bet: Indeterminacy in Poker

One of the most common pieces of poker advice given to beginners is to know why you’re betting. Because it’s important to keep things simple, it’s common to see the reasons for betting being boiled down to two basic goals: getting worse hands to call, and getting better hands to fold.

This advice is given so often because it’s a big leak to be making bets which have little chance of accomplishing either purpose. For instance, two pair is usually a fairly good hand, but betting heavily with two pair on a board which makes straights and flushes possible can easily be a mistake if it’s going to cause your opponent to fold all one-pair hands, yet never push them off a set or better.

Taking this binary too literally can lead to uncreative thinking, however. While there are some situations, especially on the river, in which one may be betting entirely or almost-entirely for one reason or the other, the vast majority of bets actually have the potential to operate in both ways. On the surface, this seems like a paradox: if a bet can simultaneously make a worse hand call and a better hand fold, that implies that the opponent’s folding range contains some hands which are stronger than some of the hands in her calling range. Isn’t that irrational?

The resolution to this paradox has to do with the imperfect information inherent in poker. Obviously, we don’t know our opponent’s cards, but there are many other things we don’t know either. First of all, prior to the river, we don’t have complete information about the strength of our own hand – we don’t, for instance, know if we will hit a flush draw, or improve our one pair to two pair or trips; plus, each of the hands in our opponent’s range could likewise either improve or not. Secondly, we don’t have full knowledge of our opponent; we may feel we have a read on him, but that read may be incorrect, and the fewer hands we’ve played with him, the more uncertainty there is in predicting his strategy. Finally, games of imperfect information tend to lead to mixed strategies, so even a fully-understood opponent will be capable of behaving unpredictably, especially with hands towards the middle of her range.

Quantum mechanics and poker

As someone with a background in physics, the imperfect information in poker and the way players cope with it has always reminded me of the revolution in scientific thinking that took place around the turn of the 20th century.

From the time of Newton to the early 1900s, the science community was fixated on the notion of determism. The idea was that if you fully understood both the initial conditions of a situation and the laws of physics, you would be able to predict the outcome with complete certainty.

This is fairly close to the truth for large-scale systems under normal conditions. Newtonian physicists were able to predict where cannonballs would land, and eventually engineers were able to produce cars, airplanes, power plants and all of the rest of the things we now enjoy, according to the seemingly rigorous and reliable laws that classical physics had set out for them.

Many lay people still think of science this way, but as physics has advanced, what physicists have been forced to recognize is that the universe, at the smallest scales, is fundamentally uncertain; not entirely random, but not entirely deterministic either, based on probabilistic distributions rather than fixed values.

Imagine that you have a situation in which a particle is seen doing Thing A 30% of the time and Thing B 70%. Under classical physics, the belief was that the particle was always destined to do one or the other, but limitations on our ability to measure the initial conditions (the particle’s position and momentum, say, among other things) were preventing us from predicting its behavior exactly. Quantum mechanics has forced us to recognize that this way of thinking is incorrect in a couple of different ways.

First of all, it’s actually impossible to measure the initial conditions with sufficient precision, without changing them in the process. Put simply, you can’t take a close look at the particle without “bumping” it in some way, so if you try to figure out exactly where it is, you’re going to change its momentum and vice versa. This is somewhat analogous to trying to get a read on an opponent who is aware that you’re doing so, and trying to adjust her strategy in response.

Secondly, we’ve realized that a particle which could do Thing A or Thing B is not really one or the other, but actually a “superposition” of both, until such time as we measure the final result and cause it to “collapse” into one state or the other. This is what Erwin Schrödinger was getting at with his famous thought experiment regarding a cat in a sealed box with a cyanide capsule which may or may not have been released. Schrödinger’s Cat is both alive and dead until the box is opened, whereupon it becomes one or the other. Similarly, we’re not usually playing against either an opponent who will call the river with second pair, or an opponent who won’t do so, but rather a superposition of those two opponents; that is, an opponent who might call is equivalent to a superposition of the opponent who will and the opponent who won’t, only becoming entirely one or the other at the moment that he decides between the two actions.

Semibluffs and protection bets

The most obvious way in which a single bet can be both for value and as a bluff comes early in the hand, when there are cards left to come. If we’re betting with a flush draw, we are betting with a hand which is a superposition of a very strong one (a made flush) and a very weak one (the missed draw), the former of which is going for value and the latter of which is bluffing; hence the term “semibluff.” Similarly, if we’re betting a marginal made hand like middle pair “for protection,” we are simultaneously both going for value and bluffing against a wide variety of hands each of which are themselves superpositions of ones which will eventually improve to beat us and ones which won’t.

Maybe you don’t like thinking in these terms. It might be more comfortable to talk about equity rather than superpositions and to consider a bet to be a bluff if it causes a hand with correct odds to fold, and a value bet if it causes a hand with incorrect odds to call. Even thinking about things this way, semibluff and protection situations tend to cause some bets to operate both ways.

Consider a situation in Hold’em where the board is something like J83, two suits, where we hold A2 with the nut flush draw. If we bet this hand as a semibluff, we are likely to get folds from many hands which are slightly ahead of us: most middle and bottom pairs, pocket pairs 7 and less, and worse Ace-highs with no draw. We’re also likely to get calls from many hands which are pretty far behind us, like T9 and worse flush draws. Thus, even when we simplify hands by considering them in terms of their equity rather than as superpositions of all possible turns and rivers, a single opponent’s range can easily contain both worse hands which he will call with incorrectly and better hands which he will fold incorrectly.

This is because we know our specific hand, while he is putting us on a range, and our respective assessments of his equity are therefore different: from his perspective, his small flush draw has better equity than bottom pair, because he’s playing against a superposition of all the hands in our range, not just the nut flush draw specifically.

The unknown opponent

It’s relatively easy to understand how bets can be “fuzzy” early in the hand, when everyone’s ranges are wide and there are cards left to come. That’s a situation that feels inherently uncertain, so I suspect that most players with a little experience are pretty comfortable with the idea that their bets can effective in multiple ways at once. But what about on the river?

On the river, it’s very tempting to think of the opponent’s range as a linear scale of hands, with our own hand being placed somewhere on that scale. After all, with no cards left to come, every hand is winning, losing or chopping with any other hand, with no uncertainty in the outcome once you know the cards. This is objectively true, but it tends to lead to the incorrect assumption that our opponent has a specific calling threshold. Often this is not the case, but even if a given opponent’s strategy is in fact to call 100% of the time with hands of a given rank and up and fold the rest, we can’t know that this is the case, or what that threshold is.

Anyone who has played enough poker has had the experience of inadvertently “bluffing for value.” Perhaps you miss your nut flush draw and fire a big bluff, only to get hero-called by a worse Ace high, or you decide to turn a small pocket pair into a bluff only to be called by the next pair down. The converse also happens, though you’re less likely to find out about it. You might go for thin value, thinking your opponent has a tendency to pay off, only to have them to read you for strength and fold a better hand. Unless they show, you’ll never know that this is what happened, but that doesn’t mean it’s not going on; we see it sometimes in televised or live-streamed final tables, where players’ hole cards are shown.

Players can of course be looser or tighter than one another, and until we have a huge number of hands on someone, we don’t know quite how they play, even if they’re not adjusting as they go. Furthermore, it’s usually going to be the case that a player who has a looser overall range on the river is going to have a lower calling threshold than a player with a tighter one. The same population of players can contain some people who will have a wide range of hands on a given river and call with any pair, and others who will only have top pair or better, but fold anything worse than two pair. If you don’t know how your opponent plays, the same bet with middle pair could have potential to get value if they turn out to be extremely loose or to bluff them if they turn out to be extremely tight.

That doesn’t mean that we should be mid-range hands blindly against unknowns; checking mid-range hands on the river is often better, but there may be times when betting works well. For instance, we may have an opponent that we know to be tight in most spots, but we’re not sure whether he likes to bluff-catch on missed flush-draw boards. If we consider him to be a superposition of an opponent who is always tight on the river regardless of board texture, and an opponent who is ordinarily tight but will call much lighter when missed draws are possible, then there may be spots where it makes sense to merge our range and bet second pair even if we’re not sure whether we’re more likely to pull off a bluff or get thin value.

Mixed strategies

Furthermore, good players play mixed strategies. Whether they’re actively trying to randomize or merely “making reads” and “changing gears,” an experienced poker player cannot always be relied upon to do the same thing in the same situation. In this way, even a well-understood strong opponent can be looked at as a superposition of several different opponents who all have different ranges and different calling thresholds. If you hold pocket 8s on a KK973 river and they’re calling 75% of the time with a 9, 50% of the time with a 7 and 20% of the time with a 3, then if you’re considering betting your eights, you have to think about both the possibility of bluffing him off of a 9, and the value that you’re getting from those times you get called by worse.

Again, the reality is that in many cases, checking with marginal hands is the way to go, but in cases where it seems like bluffing might almost but not quite be better than checking, it may be that the hidden value in occasionally getting hero-called by worse is enough to tip the balance. This is probably one of the reasons that the better AI bots like Claudico make moves which seem weird to us; it’s pretty overwhelming to put your opponent both on a range of hands and a range of responses with each, so to keep things simple, humans have a tendency to focus on our opponents’ most probable responses, whereas crunching the numbers is much easier for a computer (especially as the computation is all done in advance for bots like Claudico and stored in a table covering all possible scenarios, rather than being calculated in real-time).

A footnote about hi-lo games

Although indeterminacy in poker is something I think about a lot, the final impetus to write this article came from a Pot-Limit Omaha Hi/Lo hand posted to Twitter by Andrew “foucault82” Brokos. Although every poker variant has the potential for this kind of “Schrödinger’s bet,” it comes up all the time in Hi/Lo games, because it’s common to be either strong for one end of the pot and weak for the other, or to be marginal for both ends. This can lead to situations where it’s clearly correct to make a big bet without even knowing whether you’re going for value or as a bluff.

I was originally going to analyze Andrew’s hand, but it’s a bit complicated and this article is now running rather long, so let’s take a much simpler situation. It’s the river in a game of Pot-Limit Omaha Hi/Lo and our opponent checks to us in position. We’ve got a very strong high hand and no low, on a board where a low is possible for our opponent. If we go to showdown, we’re therefore almost always scooping if our opponent doesn’t have a qualifying low, and almost always chopping if he does – very rarely are we getting scooped or quartered.

In this situation, it makes sense to bet and probably to bet pretty big, even if we have no idea whether our opponent has a good high, a good low, neither or both. Our bet works effectively as a bluff against most of his low-only hands, as well as those which are marginal in both directions – he may even fold the nut low if the bet is large enough, and if he has no high equity and fears getting quartered. Meanwhile, it can also work out well as a value bet against his high-only hands, getting worse highs to call if he believes we’re probably on a nut low and trying to scare him out of a chop. Furthermore, so long as we’re unlikely to be beaten or tied for high, the worst case scenario is that he calls with a nut low and a slightly worse high and we chop.

The zero-downside makes this bet a pretty trivial one to make, but what’s important and interesting to realize about it is that it’s still effective even if both we and the opponent understand what’s going on; we both may know that some of his hands are going to be splitting with us while others will be getting scooped, but as long as neither of us knows which it is, we don’t care whether we’re actually bluffing or going for value, and it is the opponent who is left having to guess. This can be true even if we’re not a lock for one end or the other; in the case of Andrew’s hand, he has some equity both ways, but gets called by an opponent who is slightly worse in both directions and thinks he’s getting bullied out of a split. Whatever our actual hand, the fact that there are two halves to the pot means that very many bets in hi/lo games have both “bluff nature” and “value nature” at once.

Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Montreal, Quebec, Canada.

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